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The dependence of critical scattering on crystal lattice structure
Volume 43A, number 3
12 March 1973
PHYSICS LETTERS
THE DEPENDENCE OF CRITICAL SCATTERING ON CRYSTAL LATTICE STRUCTURE J. KOCIfiSKI Instituteof Physics, Warsaw Technical University, Warszawa, Koszykowa 75, Poland
and L. WOJTCZAK Institute of Physics, University of tbdi,
tbdi,
Poland
Received 16 January 1973 Critical scattering in monocrystals undergoing second order phase transition in particular in ferroelectrics is shown to depend on the lattice structure.
The experimental data on critical scattering of Xrays in NaN03 [ 11, which undergoes a nonferroelectric second order phase transition suggest that the explanation of critical scattering in ferroelectrics should not be tied up with the effect of spontaneous polarization. This conclusion follows from the fact that the constant scattered intensity curves drawn in the reciprocal lattice space have the same character for the ferroelectric TGS [2] as well as for the nonferroelectric NaN03. The starting point for the present note is the assumption that critical scattering in ferroelectric like TGS or BaTi03, or nonferroelectric like NaN03 crystals, is caused by the appearance in the equilibrium phase for temperatures above T, of fluctuations corresponding to the equilibrium phase below T,, that means with the fluctuations in the positions of atoms. This assumption brings us to the level of Landau’s theory of critical scattering of X-rays in crystals [3]. However, the consequences which will be drawn from this type of theory are new. Let us, following Landau, represent the denstities of the two equilibrium phases below and above Tc by p1 and p. correspondingly, and by P=Pu+VP1
(1)
the actual density above or at T,, where npl represents the fluctuation of the phase “1” in the phase “0” and 17is a parameter, which in general depends on position and time. The cross section for critical scat-
tering given in the elastic approximation by the expression: do Eq=k-
s (r)(O) r)(r)) exp (is *r) dt, k, - 2nr
(2)
contains the static correlation function (n(O) q(r) ) of the parameter n(r, t). This correlation is determined by the mean spatial distribution of the fluctuating parameter 7).In turn the mean spatial distribution of n is determined by the variation of the increase in the Gibbs potential G due to the fluctuation in 0. This potential is taken in the form analogous to that, earlier derived for ferromagnets [4] : AG = aq2 + bv4 + c(grad o)~ + dg2(grad n)2
(3)
(where a, b, c and d are phenomenological coefficients), being at the same time an extension of the formulae of Smoluchowski [S] and Landau [3]. The nonlinear differential equation for 9 which follows from the variation of (3) reduces outside the immediate vicinity of the critical point to the linear Ornstein-Zernike equation [4], which for cubic or orthogonal lattices is solved by the function r)(x, Y9z) = A exp 1-K 1(crlxl+PlyI +rlzl)] , (w2tp2ty2=
1,
(4)
with (Y,p and 7 being connected with the lattice constants. The cross section following from it has been 215
Volume 43A, number 3
PHYSICS LETTERS
first discussed for ferromagnets [6, 71. The constant intensity curves drawn in the reciprocal space which follow from (4) are compared with experiment in [8]. We note that the difference in the shape of the experimentally found constant intensity curves for NaNO, [l],andTGS [2] is easily explained in the terms of the correlation (4) since the degree of anisotropy of the cross section depends strongly on the ratio q/~ 1, that means at constant q it diminishes with growing temperature. We also point out that the constant intensity curves obtained by neutron scattering in BaTiO, [9] agree well with those following from our cross section for QI= /3 = y = 4 &, which corresponds to the fact that the tetragonal phase in BaTiO, is very near to cubic. The constant intensity curves represent a convenient means for determining the interval of temperatures where K~ ceases to determine the range of correlation, since if K: iS of the form (T-T,)“, it follows from our cross section that the area within a constant intensity curve should diminish towards zero when the critical point is approached.
216
12 March 1973
The anisotropy of the cross section is a purely crystalline effect and it should appear in all the systems having crystalline structure in which critical scattering occurs, in particular in ferromagnets as it has been earlier pointed out [6]. The various types of interactions between the atoms in these systems determine only the range of correlation. References [l] Y. Yamada, Y. Fujii and H. Terauchi, J. Phys. Sot. Japan 28 (1970) 274. [2] Y. Fujii and Y. Yamada, J. Phys. Sot. Japan 30 (1971) 1676. (31 L.D. Landau, Phys. Zs. Soviet. 12 (1937) 123. [4] J. Kociliski, L. Wojtczak and B. Mrygori, Phys. Lett. 36A (1971) 171; Acta Phys. Polon., to be published. [SM. Smoluchowski, Ann, Physik 25 (1908) 205. [6] L. Wojtczak and J. Kocitiski, Phys. Lett. 32A (1970) 389. [ 71 J. Kociriski, Lectures in the International School on Magnetism of metals in Zakopane 1970 (Warsaw 1971). [8] B. Pura and J. Przedmojski, Phys. Lett. 43A (1973) 000. [9] Y. Yamada, G. Shirane and A. Linz, Phys. Rev. 177 (1969) 848.
12 March 1973
PHYSICS LETTERS
THE DEPENDENCE OF CRITICAL SCATTERING ON CRYSTAL LATTICE STRUCTURE J. KOCIfiSKI Instituteof Physics, Warsaw Technical University, Warszawa, Koszykowa 75, Poland
and L. WOJTCZAK Institute of Physics, University of tbdi,
tbdi,
Poland
Received 16 January 1973 Critical scattering in monocrystals undergoing second order phase transition in particular in ferroelectrics is shown to depend on the lattice structure.
The experimental data on critical scattering of Xrays in NaN03 [ 11, which undergoes a nonferroelectric second order phase transition suggest that the explanation of critical scattering in ferroelectrics should not be tied up with the effect of spontaneous polarization. This conclusion follows from the fact that the constant scattered intensity curves drawn in the reciprocal lattice space have the same character for the ferroelectric TGS [2] as well as for the nonferroelectric NaN03. The starting point for the present note is the assumption that critical scattering in ferroelectric like TGS or BaTi03, or nonferroelectric like NaN03 crystals, is caused by the appearance in the equilibrium phase for temperatures above T, of fluctuations corresponding to the equilibrium phase below T,, that means with the fluctuations in the positions of atoms. This assumption brings us to the level of Landau’s theory of critical scattering of X-rays in crystals [3]. However, the consequences which will be drawn from this type of theory are new. Let us, following Landau, represent the denstities of the two equilibrium phases below and above Tc by p1 and p. correspondingly, and by P=Pu+VP1
(1)
the actual density above or at T,, where npl represents the fluctuation of the phase “1” in the phase “0” and 17is a parameter, which in general depends on position and time. The cross section for critical scat-
tering given in the elastic approximation by the expression: do Eq=k-
s (r)(O) r)(r)) exp (is *r) dt, k, - 2nr
(2)
contains the static correlation function (n(O) q(r) ) of the parameter n(r, t). This correlation is determined by the mean spatial distribution of the fluctuating parameter 7).In turn the mean spatial distribution of n is determined by the variation of the increase in the Gibbs potential G due to the fluctuation in 0. This potential is taken in the form analogous to that, earlier derived for ferromagnets [4] : AG = aq2 + bv4 + c(grad o)~ + dg2(grad n)2
(3)
(where a, b, c and d are phenomenological coefficients), being at the same time an extension of the formulae of Smoluchowski [S] and Landau [3]. The nonlinear differential equation for 9 which follows from the variation of (3) reduces outside the immediate vicinity of the critical point to the linear Ornstein-Zernike equation [4], which for cubic or orthogonal lattices is solved by the function r)(x, Y9z) = A exp 1-K 1(crlxl+PlyI +rlzl)] , (w2tp2ty2=
1,
(4)
with (Y,p and 7 being connected with the lattice constants. The cross section following from it has been 215
Volume 43A, number 3
PHYSICS LETTERS
first discussed for ferromagnets [6, 71. The constant intensity curves drawn in the reciprocal space which follow from (4) are compared with experiment in [8]. We note that the difference in the shape of the experimentally found constant intensity curves for NaNO, [l],andTGS [2] is easily explained in the terms of the correlation (4) since the degree of anisotropy of the cross section depends strongly on the ratio q/~ 1, that means at constant q it diminishes with growing temperature. We also point out that the constant intensity curves obtained by neutron scattering in BaTiO, [9] agree well with those following from our cross section for QI= /3 = y = 4 &, which corresponds to the fact that the tetragonal phase in BaTiO, is very near to cubic. The constant intensity curves represent a convenient means for determining the interval of temperatures where K~ ceases to determine the range of correlation, since if K: iS of the form (T-T,)“, it follows from our cross section that the area within a constant intensity curve should diminish towards zero when the critical point is approached.
216
12 March 1973
The anisotropy of the cross section is a purely crystalline effect and it should appear in all the systems having crystalline structure in which critical scattering occurs, in particular in ferromagnets as it has been earlier pointed out [6]. The various types of interactions between the atoms in these systems determine only the range of correlation. References [l] Y. Yamada, Y. Fujii and H. Terauchi, J. Phys. Sot. Japan 28 (1970) 274. [2] Y. Fujii and Y. Yamada, J. Phys. Sot. Japan 30 (1971) 1676. (31 L.D. Landau, Phys. Zs. Soviet. 12 (1937) 123. [4] J. Kociliski, L. Wojtczak and B. Mrygori, Phys. Lett. 36A (1971) 171; Acta Phys. Polon., to be published. [SM. Smoluchowski, Ann, Physik 25 (1908) 205. [6] L. Wojtczak and J. Kocitiski, Phys. Lett. 32A (1970) 389. [ 71 J. Kociriski, Lectures in the International School on Magnetism of metals in Zakopane 1970 (Warsaw 1971). [8] B. Pura and J. Przedmojski, Phys. Lett. 43A (1973) 000. [9] Y. Yamada, G. Shirane and A. Linz, Phys. Rev. 177 (1969) 848.